Simplify (2^-2)^(3/2)
The given problem is a mathematical expression that involves exponents and requires simplification. Specifically, it asks you to apply the rules of exponents to simplify the expression (2^-2)^(3/2), which is a power raised to another power. This involves manipulating the base and the exponent according to the laws of exponents to rewrite the expression in a simpler or more standard form.
$\left(\left(\right. 2^{- 2} \left.\right)\right)^{\frac{3}{2}}$
Answer
Solution:
Step 1: Exponent Multiplication
Combine the exponents in $\left(2^{-2}\right)^{\frac{3}{2}}$ by multiplying them together.
Step 1.1: Apply the Power to a Power Rule
Utilize the rule $\left(a^{m}\right)^{n} = a^{mn}$ to simplify the expression to $2^{-2 \cdot \frac{3}{2}}$.
Step 1.2: Simplify the Exponent
Identify and simplify common factors in the exponent.
Step 1.2.1: Extract the Factor of 2
Extract the factor of 2 from the exponent to get $2^{2 \cdot (-1) \cdot \frac{3}{2}}$.
Step 1.2.2: Simplify the Common Factors
Simplify the common factors to get $2^{\cancel{2} \cdot -1 \cdot \frac{3}{\cancel{2}}}$.
Step 1.2.3: Finalize the Exponent
Finalize the simplification of the expression to $2^{-1 \cdot 3}$.
Step 1.3: Complete the Multiplication
Complete the multiplication of the exponents to get $2^{-3}$.
Step 2: Apply the Negative Exponent Rule
Convert the negative exponent to a fraction using the rule $b^{-n} = \frac{1}{b^{n}}$ to obtain $\frac{1}{2^{3}}$.
Step 3: Calculate the Power of 2
Evaluate the power of 2 to get $\frac{1}{8}$.
Step 4: Present the Final Result
The simplified expression can be presented in various forms.
Exact Form: $\frac{1}{8}$ Decimal Form: $0.125$
Knowledge Notes:
To solve the given problem, several mathematical rules and properties are applied:
Power to a Power Rule: When you raise a power to another power, you multiply the exponents. The general form of this rule is $(a^{m})^{n} = a^{mn}$.
Simplifying Exponents: When simplifying expressions with exponents, look for common factors that can be canceled out to simplify the calculation.
Negative Exponent Rule: A negative exponent indicates that the base is on the wrong side of a fraction line. To make the exponent positive, you can take the reciprocal of the base and then raise it to the absolute value of the exponent. The rule is $b^{-n} = \frac{1}{b^{n}}$.
Evaluating Powers: To evaluate a power such as $2^{3}$, you multiply the base (2) by itself as many times as the exponent indicates (3 times in this case), which equals $2 \cdot 2 \cdot 2 = 8$.
Exact vs Decimal Form: The exact form of a number is the form that represents the value without any approximation, while the decimal form is a numerical representation that may be rounded or truncated.
In this problem, we applied the power to a power rule to combine the exponents, simplified the resulting expression, used the negative exponent rule to rewrite the expression as a fraction, and finally evaluated the power of 2 to find the exact and decimal form of the result.
